Geoscience Reference
In-Depth Information
Multiplying each equation of this system by
Q
n
(
x
) and integrating the ob-
tained equalities over
z
from0to
l
z
, we obtain equations for mode amplitudes
a
n
,b
n
,and
c
n
. In the 1D-box model (
c
A
=
c
A
(
x
)) oscillations of field-lines
do not interact with one another, and the bi-orthogonality condition (5.11) is
fulfilled. Then the infinite system of coupled equations is split into uncoupled
finite-dimensional equations of the form
j
(
d
)
k
A
(
x
)
k
n
a
n
(
x
)=
ik
y
b
n
(
x
)
4
π
c
xn
B
0
,
−
−
(5.26)
j
(
d
)
b
n
(
x
)=
k
A
(
x
)
k
n
c
n
(
x
)
4
π
c
yn
B
0
,
−
−
(5.27)
c
n
(
x
)=
−
ik
y
a
n
(
x
)
−
b
n
(
x
)
,
(5.28)
where
j
(
d
)
n
j
(
d
n
,Q
n
(
z
)
=
.
Consider the one-mode propagation. We have from (5.26)
ik
y
b
n
(
x
)
.
j
(
d
)
1
k
A
(
x
)
4
π
c
xn
B
0
a
n
(
x
)=
−
−
k
n
Substituting it into (5.27) and (5.28) yields
j
(
d
)
b
n
(
x
)=
k
A
(
x
)
k
n
c
n
(
x
)
4
π
c
yn
B
0
,
−
−
(5.29)
c
n
(
x
)=
b
n
(
x
)+
ik
y
4
π
c
j
(
d
)
k
y
k
A
(
x
)
1
k
A
(
x
)
xn
B
0
.
−
1+
(5.30)
−
k
n
−
k
n
These equations differ from (4.43)-(4.45) only in
k
being replaced by
k
n
and in the explicit account taken of the field sources. Therefore, many results
obtained for unbounded plasma in Chapter 4 can be transferred to the box
model by simply replacing the continuously changing wavenumber
k
by the
quantized wavenumber
k
n
. The coordinate
x
n
of a resonance field-lines of the
n
-th harmonics is found from
k
A
(
x
)=
±
k
n
.
(5.31)
In the non-dissipative case, the wavenumber
k
n
=
nπ/l
z
is real and (5.31)
canhavearealroot
x
n
corresponding to the resonant point. The point is a
regular singularity of (5.29)-(5.30).
If dissipation is not negligible, then
k
n
=
nπ
l
z
+
iκ
n
is a complex value. In our simple model
k
n
is independent of the
x
-coordinate.
With dissipation taken into account, (5.31) cannot be satisfied at real
x
and
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